Solution 2.3:7c
From Förberedande kurs i matematik 1
If we complete the square
\displaystyle x^{2}+x+1=\left( x+\frac{1}{2} \right)^{2}-\left( \frac{1}{2} \right)^{2}+1=\left( x+\frac{1}{2} \right)^{2}+\frac{3}{4}
we see on the right-hand side that we can make the expression arbitrarily large simply by choosing
\displaystyle x+\frac{1}{2}
sufficiently large. Hence, there is no maximum value.
